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1 Introduction Perfect Images of Generalized Ordered Spaces by Gary Gruenhage, Auburn University, Auburn, AL 36849 and David Lutzer, College of William and Mary, Williamsburg, VA 23187 Abstract. We study the class of perfect images of generalized ordered (GO) spaces, which we de- note by PIGO. Mary Ellen Rudin's celebrated result characterizing compact monotonically normal spaces as the continuous images of compact linearly ordered spaces implies that every space with a monotonically normal compactification is in PIGO. But PIGO is wider: every metrizable space is in the class, but not every metrizable space has a monotonically normal compactification. On the other hand, a locally compact space is in PIGO if and only if it has a monotonically normal com- pactification. We answer a question of Bennett and Lutzer that asked whether a (semi)stratifiable space with a monotonically normal compactification must be metrizable by showing that any semi- stratifiable member of PIGO is metrizable. This also shows that there are monotonically normal spaces which are not in PIGO. We investigate cardinal functions in PIGO, and in particular show that if K is a compact subset of a space X in PIGO, then the character of K in X equals the pseudo-character of K in X. We show that the product of two non-discrete spaces in PIGO is not in PIGO unless both are metrizable or neither one contains a countable set with a limit point. Finally, we look at the narrower class of perfect images of linearly ordered spaces, which we denote by PILOTS. Every metrizable space is in PILOTS, and if a space in PILOTS has a Gδ-diagonal, then it must be metrizable. Thus familiar GO-spaces such as the Sorgenfrey line and Michael line are in PIGO but not in PILOTS. AMS Classification Numbers Primary = 54D15; Secondary = 54D35, 54C10, 54E20, 54E35, 54F05 Key Words and phrases monotonically normal, monotonically normal compactification, PIGO, perfect image of GO-space, cardinal invariants, character, pseudo-character, metrizability, dense metrizable subspace, weaker metric topology, paracompact, D-space, semistratifiable space, β − space, Gδ-diagonal, perfect space, σ-discrete dense subset, Fpp-space, product spaces, PILOTS, perfect image of LOTS, Lindel¨ofat infinity. 1 Introduction Mary Ellen Rudin proved in [32] that every compact monotonically normal space is a continuous image of some compact linearly ordered topological space. An easy consequence of her theorem is that any space with a monotonically normal compactification is the perfect image of a generalized ordered (GO) space, and this began to be used as a tool by several researchers (see [9],[14],and [26]) to investigate the class MNC of spaces that have monotonically normal compactifications. This suggests that it might be useful to study the wider class of perfect images of GO-spaces, which we denote by PIGO, for its own sake. PIGO is a strictly larger class of spaces than MNC; for example, 1 PIGO contains the class of metrizable spaces, while MNC does not. On the other hand, we will also see that PIGO is a much narrower class than the class of monotonically normal spaces in general. One of the main results of this paper answers the main question left open in Bennett and Lutzer's study [9] of semi-stratifiable spaces with monotonically normal compactifications. They had asked whether every (semi)stratifiable space with a monotonically normal compactification must be metrizable. We give a positive answer to this question by showing that every semi-stratifiable space in PIGO is metrizable. Note that this shows that any nonmetrizable stratifiable space is an example of a monotonically normal space which is not in PIGO. Our paper is organized as follows. Section 2 contains definitions and background results. In Section 3 we show how the class PIGO is related to other familiar classes. In Section 4 we study var- ious cardinal functions in the class PIGO; for example, we show that character and pseudocharacter agree for spaces in PIGO (but not more generally for all monotonically normal spaces). Section 5 studies paracompactness and metrization of spaces in PIGO; in particular we prove there the aforementioned result about semi-stratifiable spaces. In Section 6 we show that the product of two nondiscrete spaces in PIGO is not in PIGO unless both are metrizable or neither contains a countable set with a limit point. Section 7 introduces a special subclass of PIGO that we call PILOTS, namely spaces that are perfect images of linearly ordered topological spaces. It follows from classical results of Herrlich and Morita that every metrizable space is in PILOTS. We show that if a space in PILOTS has a Gδ-diagonal, then it must be metrizable. It follows that familiar GO-spaces like the Sorgenfrey line and the Michael line are examples of spaces that are in PIGO but not in PILOTS. We also prove that any locally compact space in PIGO (equivalently, every locally compact space with a monotonically normal compactification) must be in PILOTS. Section 8 of our paper lists some open questions. 2 Definitions and background results A generalized ordered space (GO-space) is a triple (X; σ; ≤), where ≤ is a linear order on X and σ is a Hausdorff topology on X having a base of order-convex sets. If σ coincides with the open-interval topology induced by ≤, then X is a linearly ordered topological space (LOTS). A space X is monotonically normal if for each pair (H; K) of disjoint closed sets, one can assign an open set U(H; K) satisfying: (a) H ⊂ U(H; K) ⊂ U(H; K) ⊂ X n K; (b) If H ⊂ H0 and K ⊃ K0, then U(H; K) ⊂ U(H0;K0). Metrizable spaces, and more generally stratifiable spaces, are monotonically normal, as are GO- spaces. Monotone normality is a hereditary property which is preserved under closed continuous images, and it implies collectionwise-normality [22]. The following is an equivalent condition for monotone normality of a space X: (c) For each pair (p; U) where p 2 U ⊂ X and U is open, one can assign an open set V (p; U) such that p 2 V (p; U) ⊂ U, V (p; U 0) ⊂ V (p; U) whenever U 0 ⊂ U, and V (p; X − fqg) \ V (q; X − fpg) = ; if p =6 q. 2 The space X is said to be acyclically monotonically normal if it satisfies (c) above with the ≥ Tadded condition that whenever x0; x1; :::; xn is a finite sequence of distinct points with n 2, then n − f g ; i=0 V (xi;X xi+1 ) = (taking xn+1 = x0). Moody and Roscoe [31] showed that the familiar examples of monotonically normal spaces (stratifiable spaces, elastic spaces, GO-spaces) are all acyclically monotonically normal, as is any closed continuous image of an acyclically monotonically normal space. They also showed that any acyclic monotonically normal space has the Kuratowski property that van Douwen called K0. A space X is utterly normal [15] if it has a magnetic base system, i.e., a family of neighborhood bases B(x) for each x 2 X such that, if Bx 2 B(x), By 2 B(y), and Bx \ By =6 ;, then either x 2 By or y 2 Bx. (Members of B(x) might not be open, but must have x in their interiors.) A space X is UNO (Utterly Normal with respect to Open sets) if X has a magnetic base system consisting of open sets. Every utterly normal space is monotonically normal, but it is not known if the converse is true. It is shown in [14] that every space which has a monotonically normal compactification (which includes all GO-spaces) is UNO. 2 A space X is said to have a Gδ-diagonal if the diagonal f(x; x): x 2 Xg is a Gδ-set in X . We often make use the following well-known characterization: X has a Gδ-diagonal iff there is a G G 2 Tsequence (called a Gδ-diagonal sequence)S 0; 1;::: of open covers of X such that for each x X, G f g U f 2 U 2 g n2! st(x; n) = x (where st(x; ) = U : x U ). We will also use the higher cardinal G version: if µ is an infinite cardinal,T then X has a Gµ-diagonal iff there are open covers α, α < µ, of 2 G f g X such that for each x X, α<µ st(x; α) = x . The diagonal degree of X is the least cardinal µ such that X has a Gµ-diagonal. A T3-space X is said to be semi-stratifiableS if to each open set U one may assign a sequence ⊂ ⊂ Sn(U) of closed subsets of U suchS that U = n2! Sn(U), and U V implies Sn(U) Sn(V ); X is stratifiable if in addition U = n2! Int(Sn(U)) for each open U. It is well-known that every metrizable space is stratifiable, and obviously every stratifiable space is semi-stratifiable. All stratifiable spaces are monotonically normal, and X is stratifiable iff X is semi-stratifiable and monotonically normal. We will be using the nontrivial fact due to R. Heath [21] that every stratifiable space has a σ-discrete network, where N is a network for X if whenever x 2 U, U open, then there is some N 2 N with x 2 N ⊂ U. A surjective map f : X ! Y is perfect if it is continuous, closed, and has compact fibers, and f is irreducible if there is no closed subset H of X such that the restricted map f H is onto.
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